He can threaten and cow a particular lieutenant, but Leadership on a miss doesn't say you have to put your guys in line (that's pack alpha), it says your rule is contested. Presumably this means a large sub faction of your holding is defined as "rebellious" for whatever ethical, religious, or popular reason, people have decided enough is enough and they showcase it in some way. Sure the Lieutenant might decide to be the spokesman, thinking he's got a chance to take over, or maybe earn concessions at least, but the fact is.... That does not change magically the attitudes of the others involved or even majority of the people, and their vocalizations, their strikes, their terrorism, is still present and a problem.
Plus when someone fails Leadership, you get to define these fracture lines in the culture of the holding. This becomes awesome, because that polarization might get shut down by force, sure--but that doesn't change the fact that people pay a cost. You can divide families down the middle, people can get killed. Maybe someone the Hard-holder likes gets lynched. Maybe someone the hard-holder likes lynches someone unforgivably.
This move also doesn't say you've lost all control, just some. So careful how far you push it. I like to build up the threat over multiple misses, and then snowball the effects through good times and bad. Just remember the move is decided who has control. If its contested, then that control is not absolute, show this--and let them force that shit back into line on their own terms. Just remember, those lines? They just don't go away, and now you've got civil chaos too you can play with.
Also you don't get to roll leadership because you want to. You roll it only at the start of the session. If they then try to manipulate someone and fail, you should immediately push a hard move that changes the scene to such an extent that repeating that move would be impossible. Take the reigns for a second. Capture someone, split people up, announce badness (the roar of the mob when the guys throws open the window and declares they have been denied). If the hard-holder then wants to threaten the life of their lieutenant... lol. let him. But remember he did it, and make sure that Lieutenant remembers too. I guess next time there wont be people talking first, they'll just come violently and hard.